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Optimization and Computational Fluid Dynamics - Department of ...

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1 Introduction 5<br />

It seems therefore appropriate to prepare an extensive work on CFD-O.<br />

For this purpose, both theoretical foundations <strong>and</strong> practical engineering applications<br />

<strong>of</strong> optimization relying on CFD evaluations should be presented in<br />

a unified framework. This is the main challenge <strong>of</strong> the present publication.<br />

In what follows the first three chapters illustrate the considered issues <strong>and</strong><br />

introduce most mathematical tools needed to tackle this problem, in particular<br />

when considering adjoint methods which are mathematically much<br />

more dem<strong>and</strong>ing. The subsequent five chapters cover a variety <strong>of</strong> specific<br />

engineering applications (aerospace, turbomachines, automotive, heat transfer,<br />

papermaking) illustrating different possibilities to carry out successfully<br />

CFD-O depending on varying requirements concerning accuracy, computing<br />

times <strong>and</strong>/or complexity. This should allow most readers to find both the<br />

needed theoretical background <strong>and</strong> examples <strong>of</strong> practical realizations very<br />

similar to her/his own problems, so that the first few steps on the long way<br />

toward a successful CFD-O should be greatly facilitated.<br />

In order to make it even easier, let us try now to illustrate the challenges<br />

<strong>of</strong> CFD-O using only basic concepts. In Fig. 1.2 the simplest possible optimization<br />

problem is considered: find the input parameter (sometimes called<br />

degree <strong>of</strong> freedom) that minimizes the objective function (OF, sometimes<br />

also called cost function) for an analytically known, smooth function. This<br />

is a case with a single parameter (or degree <strong>of</strong> freedom, the x-coordinate in<br />

Fig. 1.2) <strong>and</strong> a single objective (the y-coordinate in Fig. 1.2) with an OF<br />

involving only a single minimum. It is therefore an extremely simplified configuration<br />

almost never found in practice. It could nevertheless represent a<br />

problem found when thermodynamically optimizing a process [1], e.g., minimizing<br />

exergy loss when varying one process parameter, knowing analytically<br />

all thermodynamical properties. In such a case, the optimization could<br />

in principle be carried out by h<strong>and</strong>, computing the full functional behavior,<br />

since this will be quite trivial. A st<strong>and</strong>ard gradient-based (also called “steepest<br />

descent”) algorithm [3] would easily find the solution <strong>of</strong> this basic problem.<br />

Otherwise, any optimization technique presented in this book would also be<br />

able to identify the optimum within a few seconds <strong>of</strong> computing time.<br />

The picture presented in Fig. 1.2 is somewhat misleading, since it suggests<br />

that the full function OF(x) is known. If this would be the case, the engineer<br />

in charge could obviously identify the optimal solution at first glance without<br />

resorting to any optimization algorithm! In practice, only a discrete set <strong>of</strong><br />

points OF(xi) withi...Np will be known at the end <strong>of</strong> the process: these<br />

are the points for which a CFD-based evaluation has been requested. This<br />

issue is illustrated in Fig. 1.3 where the successive points obtained by a simple<br />

steepest-descent algorithm are shown schematically using ∗-symbols on top<br />

<strong>of</strong> the (in fact unknown) objective function represented by the solid line in<br />

the background.<br />

To further increase the relevance <strong>of</strong> the considered problem, Figure 1.4<br />

should now be considered. It involves again a single parameter (x-axis) <strong>and</strong><br />

a single objective (y-axis) for an analytically known, smooth function, but

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